The problem we observe represents a concatenation of the two well-studied problems: the linear system state estimation problem (see, e.g., [1]}, for the case of power system state estimation) and the problem of uplink signal detection in C-RAN [2]}. For the joint problem, it is straightforward to derive (and implement at a central location) the standard minimum mean-square error (MMSE) estimator, however, such a solution comes with prohibitive \(O(N^3)\) -complexity that hinders its application for large-scale systems. By exploiting inherent sparsity within both of the component problems, an approximate MMSE solution for each problem can be obtained using the tools from probabilistic graphical models, as recently investigated for both (power system) state estimation [3]} and uplink signal detection in C-RANs [4]}. In particular, an instance of the Belief-Propagation (BP) algorithm, called Gaussian BP (GBP) [5]}, can be applied to produce an exact MMSE estimate with \(O(N)\) -complexity, thus scaling the MMSE solution to large-scale system scenarios.
